Acta Polytechnica https://doi.org/10.14311/AP.2024.64.0588 Acta Polytechnica 64(6):588–604, 2024 © 2024 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague SETTLEMENT OF PILES WITH NEGATIVE SKIN FRICTION: DESIGN APPROACHES ACCORDING TO PART 3 OF EUROCODE 7 Alena Zemanová∗, Vojtěch Anderle, Jan Masopust Czech Technical University in Prague, Faculty of Civil Engineering, Department of Geotechnics, Thákurova 7, 166 29 Prague, Czech Republic ∗ corresponding author: alena.zemanova@fsv.cvut.cz Abstract. Piles offer an effective solution for supporting structures in challenging ground condi- tions. However, ground settlement around deep foundations can induce negative skin friction, adding complexity to the geotechnical analysis and requiring careful consideration in both the pile settlement evaluation and structural integrity design. The article discusses the general recommendations provided by the informative annex of the revised Eurocode 7 version for designing piles subjected to negative skin friction. Using the Fellenius unified pile design concept, we provide a practical design framework for piles subjected to negative skin friction that extends the load-settlement curve methods proposed by Masopust, representing the standard pile design approaches in the Czech Republic. For examples presented, the introduced practical approach, also suitable for manual calculations, is in very good agreement with a refined calculation providing a nonlinear load-settlement curve. Keywords: Negative skin friction, load-settlement curves, pile settlement, pile design, soil settlement, Eurocode 7. 1. Introduction 1.1. Motivation Piles can be a very effective way to provide foundations for structures. These deep foundations are used in various situations, such as high-rise buildings, bridges, or offshore structures [1–5]. Negative skin friction is a phenomenon that can occur when the soil surrounding the pile shaft settles more than the pile itself. Soil settlement around piles can occur for a variety of reasons, including consolida- tion of soft soil layers, settlement due to embankment surcharge caused by ground elevation or other loading of adjacent structures, groundwater lowering, cyclic loading, or earthquakes [6]. The pile foundation of a bridge abutment can be a typical example of a trans- port construction, where negative skin friction can occur after backfilling behind the retaining structure and subsequent consolidation settlement of the deeper soft soil layers. Soil settlement reduces the positive shaft resistance or may even cause downward drag and increase the pile loading (with negative skin friction force called drag force). Thus, the pile is subjected to an additional settlement, or the increase in force may even result in its structural failure. The response of individual piles can be determined by load tests or alternatively calculated using empiri- cal, analytical approaches, or numerical simulations. If negative skin friction significantly affects pile re- sponse, it must be properly reflected in the pile design. For this reason, any (preliminary) calculation method should be able to consider these effects. 1.2. Literature review The discussion on negative skin friction and its con- sideration in pile design has been underway in the professional community of researchers and geotechni- cal engineers for several decades. Thus, a significant number of investigations regarding negative skin fric- tion have been carried out over the past few decades. Early work on this topic focused on full-scale field tests and experimental measurements of pile settle- ment and load distribution in driven steel pipe or con- crete piles [7–11], and also for bitumen-coated piles and pile groups [12–17]. These were followed by a se- ries of centrifuge model studies [18–22] or small-scale model tests [16, 23], and more recently large-scale experiments [24]. Analytical approaches using a closed-form solu- tion [12, 25] have been developed in parallel based on the experimental findings. These simplified methods often empirically estimate the position of the neutral point or neutral plane, where the negative skin fric- tion switches to positive shaft resistance, and further determine the drag force assuming fully mobilised negative skin friction regardless of the magnitude of the soil settlement. The existing simplified solutions usually overestimate the depth of the neutral plane and thus can be considered as an upper limit of the response [24, 26]. In contrast to previous studies, Fellenius suggested that the negative skin friction problem is a pile set- tlement issue rather than an ultimate geotechnical capacity issue and introduced his unified pile design concept accounting for negative skin friction in a pile settlement analysis [27]. His iterative scheme involves balancing the forces acting on the pile and the set- 588 https://doi.org/10.14311/AP.2024.64.0588 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 64 no. 6/2024 Settlement of piles with negative skin friction tlement of the pile and soil simultaneously, as they are related and cannot be considered separately. This approach, later generalised to pile groups [6, 28], is one of the most widely used simplified methods for estimating negative skin friction. Later, continuum approaches were proposed for homogeneous soil [29], based on the theory of ho- mogeneous elastic half-space, employing Mindlin’s solution [30] and later modified for a layered soil de- posit [31] using a mean modulus. This analytical model was verified against finite element analyses from the literature, and a practical procedure was pro- posed using a parametric dimensionless solution [31]. Poulos also introduced an approximate hand method for piles subjected to axial load and negative skin fric- tion [32, 33]. Unlike Fellenius, he does not iteratively search the neutral plane but sets it at the depth of the soil movements. This plane divides the ground profile into consolidating and stable zones and designs a suf- ficient pile length to meet the design requirements, causing structural capacity and pile settlement to be affected by negative skin friction. His conservative approach provides meaningful predictions of pile be- haviour (compared to a boundary element solution) for piles subjected to relatively large soil settlements. The load-transfer method [34] has gradually become a popular tool in pile design [35–41] widely accepted in practice [42, 43]. This approach requires stress transfer functions derived from field or laboratory tests describing shear stresses acting on the pile shaft as a function of the pile settlement. A hyperbolic law is often assumed to govern the pile-soil interface behaviour [37, 44, 45]. Wong et al. proposed a load- transfer model using hyperbolic soil springs for the pile-soil interface to address the behaviour of piles under negative skin friction in stratified soil deposits and established a procedure for evaluating the input parameters from conventional soil tests [37]. In gen- eral, the load transfer method framework provides a suitable framework for analysing the nonlinear pile settlement and also for performing inverse analyses, e.g., [37, 42, 46]. In addition, the effect of the negative skin friction time-dependency due to the nonlinear soil consolidation and loading–unloading conditions can be taken into account, such as in an improved hyperbolic model using the load transfer method for the pile–soil interface in [45]. Numerical methods, often used to verify other ap- proaches, complete the list of computational meth- ods able to capture the increasing number of factors related to soil-pile interaction and negative skin fric- tion [17, 23, 26, 47–51]. The importance of properly modelling the soil-pile interaction was demonstrated in [48] using both the no-slip and slip analyses for the prediction of the dragload development. In this elastoplastic finite element analysis, an isotropic elas- tic model represented single piles and piles in groups, whereas a non-associated Mohr-Coulomb model was used for soils. Unlike the slip analysis using interface elements, the conventional no-slip continuum analy- sis predicted unrealistic responses in which the dra- gloads were substantially overestimated. As shown in a study focused on the dragload group effect for pile groups [26], the finite difference method with in- terface elements can be used equally successfully for the numerical modelling of pile groups under nega- tive skin friction. Another detailed simulation using geotechnical software coupled with programs for struc- tural analysis was introduced in [49] to analyse the response of buildings with pile foundations to a tunnel construction, where the ground surface settlements may also induce negative skin friction on the pile shaft. Recently, more and more advanced numerical models have been developed, such as the elasto-viscoplastic model taking into account the effect of soil creep [51]. As seen from the above overview, many sophisti- cated approaches were introduced. However, they are often too complicated and time-consuming for practi- cal use and require a significant amount of input data, which are often not available and must be estimated. 1.3. Negative skin friction in current European standards The recommendations in design standards, including the European standards for geotechnical structures Eurocode 7 (EC7), have gradually evolved based on the developments in the field. EC7 [52] aimed to bridge a gap between an explicit and codified approach for the upper structure above the ground level and simple or empirical calculations for the underlying geotechnical substructures based on highly subjective assessments of design properties [53]. Nevertheless, the first version of EC7 contains only a few very general recommendations for designing piles subjected to negative skin friction and does not give detailed instructions or examples for its consider- ation [54]. However, the second generation of EC7, which is currently in the approval process, extends the design guidelines on assessing negative skin friction in one of its informative annexes in the third part of the standard prEN 1997-3: Annex C.9 Downdrag due to vertical ground movements [43]. Outlines of a rigorous interaction model and simplified approaches for the downdrag consideration are briefly introduced in the annex. Let us summarise its key points: • The possible adverse effect of negative skin friction shall be included in both limit states – always in the serviceability limit state and for the drag force higher than any variable compressive forces applied to the pile also in the ultimate limit state. • A ground-pile interaction analysis shall provide the depth of the neutral plane (the boundary between negative and positive friction and the point of the equality of the ground and pile settlement) comple- mented with force and displacement distributions. Note that the position of the neutral plane differs for both limit states. 589 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica • The drag force, classified as permanent action, re- sults from negative skin friction forced by reverse relative soil-pile movement – downdrag. A cau- tious drag force estimate should be derived from the upper values of the ground properties. It is clear from the above points that the forthcoming version of the European standard EC7 deals with the negative skin friction for both limit states. As men- tioned in the literature review, Fellenius suggested that negative skin friction design should be aimed at pile settlement [27]. Concerning the ultimate limit state, higher axial force due to the negative skin fric- tion should be examined as the structural integrity of the pile itself may be affected [32] whereas the ultimate geotechnical axial load capacity of a pile is not affected by the negative skin friction (unless a strain-softening at the pile/soil interface occurs [33]). The rigorous interaction model is schematically shown in Figure 1. The ground settlement ssoil, includ- ing the immediate, primary, and possibly secondary consolidation, should consider changes in effective stresses, ground stiffness, and the depth of the com- pressible ground. For pile settlement spile, any analyt- ical, empirical, numerical, or other approach can be used, taking into account the stress distribution. It is pointed out that the pile and ground settlement and the neutral point can be found from the interaction analysis. If the compressive forces are combined in both limit states, then either the drag force or the leading variable action should be considered. For specific cases, the guidance provides the follow- ing simplified approaches: • For serviceability limit state, the base of the con- solidating soil layer can be assumed as the neutral plane, and the drag force corresponds to the full depth of the settling soil layer. • For ultimate limit state, the base of the consoli- dating soil layer can be considered as the neutral plane if the pile settlement is much smaller than the settlement of the surrounding ground. If the pile settlement is greater than that of the surrounding soil, the neutral point can be located at the ground surface, and the drag force may be disregarded. 1.4. Goal of the study This study aims to show how negative skin friction can be incorporated into standard pile design methods. The new Part 3 of EC7 (prEN 1997-3) recommen- dations motivated the approaches presented in this paper. We provide a simple framework for design- ing piles subjected to negative skin friction extending the load-settlement curve methods proposed by Ma- sopust [55–59]. We limit our attention to the service- ability limit state as it is very likely to be the decisive state for the design of large-diameter boring piles in soft soils with its base in soils or weak rocks, which are the subject of interest of this study. neutral plane negative skin friction positive shaft friction mobilised toe resistance applied load settlement spile = ssoil ssoil(z) spile(z) z Figure 1. Settlement and force distribution. 2. Methods for pile-settlement and negative skin friction consideration Practical approaches for pile settlement design are often based on the load-settlement curve method or use load transfer functions [43]. The design methods proposed by Masopust [55–58] belong to the group of engineering approaches focusing on the design of axially-loaded bored piles. Approximately 350 static load tests performed on bored single piles in the Czech Republic and partially in Germany provided the dataset for back analysis of pile responses in dif- ferent soil deposits. These field test results allowed to establish a quadratic-linear approximation of the load-settlement curve, useful for manual calculations, and a computing procedure providing a nonlinear load- settlement curve. Both approaches, used for practical pile design for decades, consider the effect of the pile in- stallation, i.e. pile drilling, borehole cleaning, method of concreting, and shaft insulation, on the bearing capacity of the pile. 2.1. Quadratic-linear load-settlement curve approximation The first method is based on the Poulos practical model for load-settlement analysis of piles [31, 58]. Likewise, in the original approach, the construction of the load-settlement curve is simplified to the de- termination of two points, corresponding to the full mobilisation of skin friction (sy, Ry) and 25 mm set- tlement (s25, R25). However, compared to the original procedure, the first part of the load-settlement curve, before the point (sy, Ry), was replaced by a parabola (Figure 2), and the method was modified using the measured responses and their back analysis for the predictions of the shaft and base resistance. Detailed instructions for the construction of the load-settlement curve can be found in Appendix A.1 and also in the Czech standard ČSN 731004 Geotechnical Design – Foundations – Requirements for calculation meth- ods [59]. According to those documents, the shaft 590 vol. 64 no. 6/2024 Settlement of piles with negative skin friction actual LSC limit LSC approximation 2◦ 1◦ Rb,25 Rsy s25 sy Ry R25 RsyβRy load settlement Figure 2. Load-settlement curve (LSC) approximation. neutral plane ground settlement settlling soil layer stable soil layer ssoil(z) ssoil(z) = 0 z Figure 3. Pile in settling and stable zone. friction and base resistance values are then derived from the tabulated values of regression coefficients. In terms of negative skin friction incorporation, this approximation method can be combined with the simplified approach mentioned in EC7, i.e. as far as a boundary between less compressible stable soil and softer settling soil can be estimated, this boundary is conservatively considered the neutral plane (Figure 3). Then, the drag force can be determined in the settling zone under the assumption that the full mobilisation of the negative skin friction occurred [33]. However, some studies suggest reducing the negative skin fric- tion compared to its positive values, and a degree of mobilisation corresponding to a factor of 0.67 can be found in the literature [27]. Also, the Czech national standard ČSN 731004 [59] notes that the negative friction can be estimated as 70 % of the positive shaft friction. Alternatively, the depth of the neutral plane can be estimated with more detailed procedures described in [27, 32]. For our comparative study, we used the Fellenuis unified pile design concept [27] combined with the semi-empirical load-settlement curve approx- imation from [58]. A set of pile-settlement plots and neutral plane pile/soil settlement load/pile-resistance spile = ssoil ssoil(z) spile(z) pile resistancez loading force Figure 4. Equilibrium of settlements and forces. i si−1 si∆li li di qsi q0 Ri−1 Ri Figure 5. Scheme of pile discretisation. corresponding load distribution curves (starting from the base resistance and increasing with the positive shaft resistance) is constructed. Then, the location of the neutral plane is found iteratively to achieve both the load/resistance and pile/soil settlement equi- librium in the system (Figure 4). Finally, the pile settlement corresponding to this point is supposed to be the resultant one. 2.2. Procedure providing nonlinear load-settlement curve In the nonlinear approach, the pile is divided into a few segments (Figure 5). A sequence of small, grad- ual growing vertical displacements (settlements s0) is given to the pile base, and the corresponding pile base resistance q0 is evaluated. Then, the shaft fric- tions qsi, axial forces acting on the bottom and top cross-section Ri−1 and Ri, elastic contractions of the pile elements ∆li, and settlements at the top of el- ement si are determined subsequently from the pile base up to the pile head [31, 34, 55, 58]. This procedure provides settlement of the pile head with the corresponding pile load force and allows the construction of the nonlinear load-settlement curve. 591 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica Example 2Example 1 clay IC = 0.50 clay IC = 0.5 weak rock R5 weak rock R5 sand ID = 0.50 h1 = 14 m h1 = 5 m h2 = 1 m h2 = 14 m h3 = 1 m Figure 6. Scheme of analysed piles in stratified soil and weak rock deposits containing compressible clay. The mobilised skin frictions qsi and pile base resis- tances q0 are related to the pile movements by rela- tionships derived, again, from the back-analyses of the static loading tests and summarised in Appendix A.2. The relative pile/soil displacements are evaluated and used in the friction-settlements relationships if the surrounding soil settles. Thus, a nonlinear stress load-settlement curve reflecting the negative skin fric- tion can be produced straightforwardly. Then, the increased pile settlement caused by the drag force can be directly retrieved from the shifted curve. 3. Examples To show the effect of negative skin friction on pile settlement, we analyse the responses of two piles pass- ing through a thick layer of soft to medium stiff clay with consistency index IC = 0.5 (Figure 6). In both examples, the pile base is located in a weak rock R5 (compressive strength 1.5–5 MPa), and for the 20 m long pile, the clay layer is covered by a medium-dense sand layer (ID = 0.5). The lengths of the bored piles are 15 m and 20 m, with the diameter of the pile shaft of 0.8 m. Both concrete piles are subjected to a vertical force, while the soil surrounding the pile is further subjected to a surface load of 100 kPa, corresponding to an em- bankment approximately 5 m high. This additional surcharge causes soil settlement determined accord- ing to the Czech standard [59] using the oedometric moduli and unit weights listed in Table 1. 4. Results and discussion From the problems outlined above, it is clear that the simplified approach mentioned in EC7 would predict enormous pile settlements in these cases because the stable soil layer surrounding the lower part of the piles is only 1 m thick. Such a low neutral plane position, close to the pile base, is very conservative and would significantly increase the force acting on the pile. The pile geometry would need to be re-designed due to the unacceptable settlement [60]. Parameter Unit Ex. 1 Ex. 2 Loading force R [kN] 1 600 1 900 Surcharge [kPa] 100 100 Bored concrete pile Elastic modulus [GPa] 30 30 Sand ID = 0.5 Unit weight [kN m−3] - 19 Oed. modulus [MPa] - 41 Clay IC = 0.5 Unit weight [kN m−3] 20 20 Oed. modulus [MPa] 32 32 Weak rock R5 Unit weight [kN m−3] 21 21 Oed. modulus [MPa] 120 120 Table 1. Loading and material properties. 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 1.63 MN, sy = 10.7 mm R = 1.60 MN, s = 10.3 mm R25 = 2060 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] neut. plane: 2.76 m 23.6 mm 24.0 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1814 kNneutral plane: 2.76 m Resistance Resistance, Ry Resistance, R Load + NSF Figure 7. Example 1 – Quadratic-linear load- settlement curve (LSC) unaffected by negative skin friction (NSF) [59]; (colour online). For this reason, let us present a more detailed model dealing with pile load-settlement curves, providing better predictions for the neutral plane positions. Ap- pendix A includes additional data to reproduce the results presented. Furthermore, a GitLab repository was also created [61] containing the source codes for the graphs in Section 4. 4.1. Example 1 The values of the vertical forces acting on the pile heads are chosen so that the pile settlements with- out the negative skin friction were around 10 mm (Figure 7). Thus, the quadratic-linear approxima- tion of the load-settlement curve for the 15 m long pile predicts a 10.3 mm settlement corresponding to the 1 600 kN vertical load, i.e. just above the point corresponding to the full mobilisation of skin friction. 592 vol. 64 no. 6/2024 Settlement of piles with negative skin friction 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 1.63 MN, sy = 10.7 mm R = 1.60 MN, s = 10.3 mm R25 = 2060 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] neut. plane: 2.76 m 23.6 mm 24.0 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1814 kNneutral plane: 2.76 m Resistance Resistance, Ry Resistance, R Load + NSF (a). Settlement. 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 1.63 MN, sy = 10.7 mm R = 1.60 MN, s = 10.3 mm R25 = 2060 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] neut. plane: 2.76 m 23.6 mm 24.0 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1814 kNneutral plane: 2.76 m Resistance Resistance, Ry Resistance, R Load + NSF (b). Force evolutions. Figure 8. Example 1 – Settlement and force evolutions including negative skin friction (NSF) effect along the pile length provided by the practical approach [59]; (colour online). To determine the influence of the negative skin fric- tion, settlement and force evolution curves (Figure 8) are constructed using the load-bearing capacities of the pile base and shaft according to the approximate method in Appendix A.1. Figure 8a shows the evolu- tion of the pile settlement along its length compared to the soil settlement induced by the additional sur- face load. A set of pile settlements is constructed, assuming that the pile head settlement gradually in- creases from zero by 5 mm and the corresponding pile base settlement is simply estimated from the head settlement using the factor Rk representing the cor- rection factor for pile compressibility, see [31] and Appendix A.1. The settlements corresponding to the prescribed load suser and to the full mobilisation of skin friction sy are highlighted. It is reasonable to assume that the settlement of the same pile affected by negative skin friction must be higher. The compar- ison with the soil settlement shows that even using the original pile settlement without the negative skin friction effects (10.3 mm) substantially increases the position of the neutral plane, from the original depth of 14 m below the level of the pile head to a depth of approximately 8 m. Let us further refine the estimation of the neutral plane’s position by balancing the equilibrium of the load force and the pile-bearing resistance on one side and the equality of settlement of the pile and the surrounding soil on the other. For this reason, the predictions of pile resistances corresponding to the previously retrieved pile settlements are determined (Figure 8b). For each value of the head settlement, the pile base resistance can be derived assuming that the relationship is linear with a known value of the base resistance at the shaft resistance activation (Ap- pendix A.1). The pile bearing resistances gradually increase from the original pile base resistances at the level of the base towards the pile head due to the increasing pile shaft resistances calculated from the positive skin friction in individual soil layers. On the contrary, the loading force increases due to the drag load in the opposite direction, starting with the user-defined loading force at the pile head and end- ing with a maximum value at the pile base when the whole shaft is theoretically subjected to negative skin friction. The additional increase in the loading force acting on the pile due to the down drag is calculated from the negative skin friction derived from the fully mobilised positive one when the degree of mobilisation is assumed to be 0.7 [54, 59]. Finally, the neutral plane’s position is found at the level where the soil-pile-settlement equality and load-resistance equilibrium coincide. For this example, this plane is located at a depth of 2.76 m measured from the pile head, and the pile head settlement cor- responds to the value of 24 mm, i.e. it has more than doubled compared to its value without the negative skin friction effects. The previous practical approach addressed how to account for negative skin friction when using the load- settlement semi-empirical approximation. The neg- ative skin friction incorporation is more straightfor- ward for the refined approach providing a nonlinear load-settlement curve. The only modification of the original procedure, presented in Appendix A.2, is that in calculating the mobilised skin friction, the pile set- tlement is replaced by its relative value, considering the surrounding soil vertical movement. The comparison of both analyses is visualised in Figure 9. First, three load-settlement curves are com- pared (Figure 9a). The grey ones correspond to the quadratic-linear approximation and to the nonlinear 593 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.60 MN, s = 13.5 mm R = 1.60 MN, s = 23.5 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] neut. pl.: 2.88 m C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] 23.513.5 24.0 neutral plane: 2.88 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1706 kNneutral plane: 2.88 m with NSF w/o NSF with NSF (APP) (a). Load-settlement curves. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.60 MN, s = 13.5 mm R = 1.60 MN, s = 23.5 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] neut. pl.: 2.88 m C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] 23.513.5 24.0 neutral plane: 2.88 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1706 kNneutral plane: 2.88 m with NSF w/o NSF with NSF (APP) (b). Mobilised skin frictions. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.60 MN, s = 13.5 mm R = 1.60 MN, s = 23.5 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] neut. pl.: 2.88 m C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] 23.513.5 24.0 neutral plane: 2.88 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1706 kNneutral plane: 2.88 m with NSF w/o NSF with NSF (APP) (c). Settlement. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.60 MN, s = 13.5 mm R = 1.60 MN, s = 23.5 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 D ep th [m ] neut. pl.: 2.88 m C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 D ep th [m ] 23.513.5 24.0 neutral plane: 2.88 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 D ep th [m ] 1706 kNneutral plane: 2.88 m with NSF w/o NSF with NSF (APP) (d). Load evolutions along the pile length. Figure 9. Example 1 – Comparison of the load-settlement curves, mobilised skin frictions (negative (NSF), positive (PSF), or 70 % fully mobilised (0.7FMSF)), and settlement and load evolutions along the pile length by the refined nonlinear approach (NL) against the practical approximation (APP) using the quadratic-linear (QL) load-settlement curve; (colour online). load-settlement curve unaffected by the negative skin friction. The black curve represents the pile behaviour considering the negative skin friction. This nonlinear curve is shifted because the pile settles due to the down drag even though a zero loading force is applied at the pile head. In addition, the mobilised skin frictions are pre- sented (Figure 9b). In the practical approximation, the limit shaft resistance is derived from the 70 % of the fully mobilised skin frictions, see Equation (4) in Appendix A.1, and we consider the same limit value for the negative skin friction. The refined approach determines the level of shaft friction based on the relative vertical movement of the pile and the adja- cent soil. Thus, the shaft friction in the area around the neutral plane does not reach its limit value (Fig- ure 9b). This reduced skin friction is then reflected in the evolution of the pile load-resistance curve along the pile length when the largest force action on the pile (Figure 9d) decreases from 1814 kN (predicted by the practical approach) to 1706 kN (predicted by the refined approach). Also, the lower negative skin friction results in a slightly smaller pile settlement; the practical approach provides a head settlement of 24 mm compared to the refined value of 23.5 mm (Figure 9c), which is on the safe side. If we compare the resulting position of the neutral plane, it remains almost unchanged for this example. 4.2. Example 2 The same analyses are repeated and performed for the second example, considering a longer pile passing through three layers of soil, where the layer of clay is additionally covered with a layer of sand (Figure 6). 594 vol. 64 no. 6/2024 Settlement of piles with negative skin friction 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 2.2 MN, sy = 13.1 mm R = 1.9 MN, s = 10.1 mm R25 = 2461 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] D05 C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] neutral plane: 3.10 m 26.9 mm 27.7 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2213 kNneutral plane: 3.10 m Resistance Resistance, Ry Resistance, R Load + NSF Figure 10. Example 2 – Quadratic-linear load- settlement curve unaffected by negative skin fric- tion [59]; (colour online). First, the behaviour of the pile without the influ- ence of the surrounding ground settlement is analysed, and the resulting load-settlement curve is shown in Figure 10. In this case, the loading forces transferred from the upper structure to the pile would cause a set- tlement of approximately 10 mm, again, above the point of the full mobilisation of skin friction. To derive the position of the neutral plane, the re- sulting settlement, and loading force, including the drag force, it is again necessary for the practical ap- proach to construct a set of curves corresponding to the pile settlements coupled with the relevant pile resistances (Figure 11). The comparison of the inter- section of pile and soil settlements and load-resistance equilibria predicts the neutral plane at 3.1 m under the pile head, the load increment of 313 kN, and the pile head settlement of 27.7 mm. Figure 11 further shows that a simplified approach, considering the neutral plane at the interface of con- solidating and stable layers, would predict the total loading force, which mostly the pile base would have to carry, to be over 3 500 kN. Considering the initial pile settlement unaffected by negative skin friction would reduce the neutral plane position to approxi- mately 12 m depth below the pile head with a loading force of 3 000 kN. Let us check the results again with the refined ap- proach using the nonlinear load-settlement curve con- struction (Figure 12). The comparison shows that the skin friction is not fully mobilised in the first five metres of sand from the pile head. The neutral plane, corresponding to the position where the negative skin friction switches to the positive one, is predicted at a depth of 3 m. The increase in the total loading force due to the down drag load is 179 kN, and the predicted settlement of 27.4 mm is again in good agreement with the previous practical analysis. 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 2.2 MN, sy = 13.1 mm R = 1.9 MN, s = 10.1 mm R25 = 2461 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] D05 C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] neutral plane: 3.10 m 26.9 mm 27.7 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2213 kNneutral plane: 3.10 m Resistance Resistance, Ry Resistance, R Load + NSF (a). Settlement. 0 1000 2000 3000 4000 5000 Load R [kN] 0 5 10 15 20 25 Se tt le m en t s [m m ] Ry = 2.2 MN, sy = 13.1 mm R = 1.9 MN, s = 10.1 mm R25 = 2461 kN LSC no NSF fully mob. skin [Ry, sy] prescribed [R, s] 0 25 50 75 100 125 150 Fully mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] D05 C050 R5 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] neutral plane: 3.10 m 26.9 mm 27.7 Pile settlement Pile set., sy Pile set., s Soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2213 kNneutral plane: 3.10 m Resistance Resistance, Ry Resistance, R Load + NSF (b). Force evolutions. Figure 11. Example 2 – Settlement and force evo- lutions including negative skin friction (NSF) effect along the pile length provided by the practical ap- proach [59]; (colour online). 5. Conclusion This study compared two approaches: (1.) a practical design framework for piles subjected to negative skin friction combining the load-settlement curve approximation proposed by Masopust with the Fellenius unified pile design concept, (2.) a refined approach relating the mobilised skin friction directly to movements of individual pile segments and providing a nonlinear relationship between the pile head settlement and pile resistance. The main conclusions from this study are: • The surrounding soil settlements resulting in neg- ative skin friction and drag force can significantly increase the pile settlement. 595 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.9 MN, s = 6.8 mm R = 1.9 MN, s = 27.4 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] n. p.: 3.00 m D05 C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] 27.46.8 27.7 neut. plane: 3.00 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2079 kNneutral plane: 3.00 m with NSF w/o NSF with NSF (APP) (a). Load-settlement curves. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.9 MN, s = 6.8 mm R = 1.9 MN, s = 27.4 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] n. p.: 3.00 m D05 C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] 27.46.8 27.7 neut. plane: 3.00 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2079 kNneutral plane: 3.00 m with NSF w/o NSF with NSF (APP) (b). Skin frictions. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.9 MN, s = 6.8 mm R = 1.9 MN, s = 27.4 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] n. p.: 3.00 m D05 C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] 27.46.8 27.7 neut. plane: 3.00 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2079 kNneutral plane: 3.00 m with NSF w/o NSF with NSF (APP) (c). Settlement. 0 1000 2000 3000 4000 Load R [kN] 0 10 20 30 40 50 Se tt le m en t s [m m ] R = 1.9 MN, s = 6.8 mm R = 1.9 MN, s = 27.4 mm NL with NSF NL w/o NSF QL w/o NSF (APP) 0 50 100 150 Mobilised skin friction [kPa] 0 5 10 15 20 D ep th [m ] n. p.: 3.00 m D05 C050 R5 PSF/NSF PSF 0.7FMSF 0 10 20 30 40 50 Pile settlement and soil settlement [mm] 0 5 10 15 20 D ep th [m ] 27.46.8 27.7 neut. plane: 3.00 m with NSF w/o NSF with NSF (APP) soil settlement 0 500 1000 1500 2000 2500 3000 Load [kN] 0 5 10 15 20 D ep th [m ] 2079 kNneutral plane: 3.00 m with NSF w/o NSF with NSF (APP) (d). Evolutions along the pile length. Figure 12. Example 2 – Comparison of the load-settlement curves, skin frictions (negative (NSF), positive (PSF), or 70 % fully mobilised (0.7FMSF)), and settlement and load evolutions along the pile length by nonlinear approach (NL) or practical approximation (APP) using quadratic-linear (QL) load-settlement curve; (colour online). • The simplified approach, mentioned in the second generation of EC7, estimating the neutral plane at the boundary of the settling and stable layer, leads to an uneconomical pile design for ground pro- files with a significant thickness of the compressible layer(s). • The examples presented show a very good agree- ment between the practical approach based on the load-settlement approximation and the response of the refined model. Let us point out that the used friction-settlements relationships from [55] or Appendix A may be further updated using load-transfer functions introduced in the proposed European standard [43] or the recent literature [42, 44, 62]. The next logical step would be to validate the model against experimental measurements. Unfortunately, a field measurement is beyond the scope of the grant projects that supported this research. Acknowledgements The financial support from the Czech Grant Agency, project No. 22-12178S, and by the Czech Tech- nical University in Prague, within the project SGS23/040/OHK1/1T/11, is gratefully acknowledged. References [1] H. G. Poulos. Tall building foundations: Design methods and applications. Innovative Infrastructure Solutions 1(1):10, 2016. https://doi.org/10.1007/s41062-016-0010-2 [2] H. G. Poulos, A. J. Davids. Foundation design for the emirates twin towers, dubai. Canadian Geotechnical Journal 42(3):716–730, 2005. https://doi.org/10.1139/t05-004 596 https://doi.org/10.1007/s41062-016-0010-2 https://doi.org/10.1139/t05-004 vol. 64 no. 6/2024 Settlement of piles with negative skin friction [3] W. F. Baker, C. Brown, J. J. Pawlikowski, D. S. Rankin. 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Masopust Acta Polytechnica Dn D1 L d0 d1 di hn h1 hi dn Di soil layer n soil layer i soil layer 1 Figure 13. Pile in a layered soil deposit Appendix A. A.1. Practical approach providing a quadratic-linear approximation of the load-settlement curve The load-settlement curve is approximated by a parabola continued by a straight line [55–58] and Figure 2. The coordinates of two points corresponding to the full mobilisation of skin friction (sy; Ry) and to the axial load-bearing capacity at the 25 mm settlement (s25 = 25 mm; R25) are necessary to construct the load-settlement curve. The values of settlement and loading force corresponding to the full mobilisation of skin friction (sy; Ry) determine the first branch of the parabola-shaped curve for the load-interval R ∈ ⟨0, Ry⟩: s = sy ( R Ry )2 . (1) In the interval starting at the full mobilisation point and ending at the load-bearing capacity of pile corresponding to the 25 mm settlement R ∈ ⟨Ry, R25⟩, the second branch of the load-settlement curve is given by the straight line with the coordinates of the endpoint (s25 = 25 mm; R25): s = sy + (s25 − sy) R − Ry R25 − Ry . (2) Thus, the increase in shaft resistance is assumed to be quadratic whereas the base resistance is supposed to be linearly related to the pile head settlement. Taking into account the notation for a single pile geometry in a layered soil deposit in Figure 13, the relations for determining all quantities needed for the load-settlement curve construction, i.e., the pile settlement corresponding to the full mobilisation of skin friction sy, the pile load corresponding to the full mobilisation of skin friction Ry, and the load-bearing capacity of the pile R25, are given by equations below. The value of the axial load-bearing capacity of the pile R25 consists of the shaft and base resistance, Rsy and Rb,25: R25 = Rsy + Rb,25. (3) The pile is divided into several parts corresponding to the independent soil layers (Figure 13), and then, the fully mobilised load-bearing capacity of the pile shaft is given by combining skin friction contributions from the soil layers: Rsy = 0.7mπ n∑ i=1 dihiqsi, (4) where the coefficient reflecting the effect of the pile shaft surface m is given in Table 2 and the diameters of pile segments in the i-th layer d i and the thicknesses of soil in the i-th layer hi are set according to Figure 13. Back analysis of static load tests on single bored piles provided a set of load-settlements curves, from which the 600 vol. 64 no. 6/2024 Settlement of piles with negative skin friction Value Methods of pile installation and insulation 1.0 Unsupported excavations, concreting in a dry borehole or under the water level 0.9 Excavations under excavation-support fluids 0.7 Piles with secondary shaft insulation made of PVC or PE foils with thickness > 0.7 mm 0.5 Piles with secondary shaft insulation made of PVC foil and wire netting, 0.15 Piles with the permanent casting of steel tubes Table 2. Installation coefficient reflecting the effect of the pile shaft surface m. Rock/Soil a b e f Weak rocks with uniaxial compressive strength of 15–20 MPa (R3) 246 226 2841 1299 Weak rocks with uniaxial compressive strength of 5–15 MPa (R4) 170 139 1616 1155 Weak rocks with uniaxial compressive strength of 1.5–5 MPa (R5) 132 95 958 704 Cohesionless soils with density index ID = 0.9 (D9) 154 116 1597 1399 Cohesionless soils with density index ID = 0.7 (D7) 91 48 490 445 Cohesionless soils with density index ID = 0.5 (D5) 62 16 268 175 Cohesive soils with consistency index IC ≥ 1.0 (C100) 97 109 988 1084 Cohesive soils with consistency index IC = 0.75 (C075) 71 65 593 617 Cohesive soils with consistency index IC = 0.5 (C050) 46 21 198 150 Table 3. Regression coefficients a, b, c, and d [kPa] for various rocks and soils. relationships for the ultimate shaft frictions qsi and the base resistance corresponding to the full mobilisation of the shaft friction were derived. Thus, the limit shaft friction in the i-th layer is provided by a hyperbolic law: qsi = a − b( Di di ) , (5) and the base resistance q0 corresponding to the full mobilisation of the shaft friction is described likewise: q0 = e − f( L d0 ) , (6) using the couples of regression coefficients a, b and e, f according to Table 3 and slenderness ratios derived from the distance of the pile cap and the middle of the i-th layer Di, the pile length L, and the diameters of pile shaft di or base d0 (Figure 13). Due to the layered soil profile, the weighted average of skin frictions along the pile shaft may be determined as: qs = n∑ i=1 dihiqsi n∑ i=1 dihi . (7) The base capacity corresponding to the 25 mm pile settlement, assuming its linear increase in depth, is given by: Rb,25 = βRy s25 sy , (8) employing the coefficient of load transfer to the pile base: β = q0 q0 + 4qs L d0 . (9) Thus, the pile load corresponding to the full mobilisation of the shaft friction may be expressed according to Poulos’s method [31] as: Ry = Rsy 1 − β . (10) Finally, the pile settlement corresponding to the full mobilisation of skin friction is determined following [31] as: sy = I Ry dEs , (11) 601 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica Layer thickness Moduli for given pile diameter and weak rock type [MPa] h d = 0.6 m d = 1.0 m d = 1.5 m [m] R3 R4 R5 R3 R4 R5 R3 R4 R5 1.5 50.3 28.2 20.0 72.3 35.0 24.7 85.5 33.5 22.3 3 64.5 43.1 30.8 106 57.3 41.0 138 58.8 41.2 5 58.2 41.3 75.3 54.8 87.9 63.7 10 87.5 61.6 115 83.2 133 97.0 Table 4. Secant deformation moduli Es [MPa] for piles in weak rocks. Layer thickness Moduli for given pile diameter and cohesionless soil type [MPa] h d = 0.6 m d = 1.0 m d = 1.5 m [m] ID = 0.5 0.7 0.9 ID = 0.5 0.7 0.9 ID = 0.5 0.7 0.9 1.5 11.0 13.7 28.3 12.8 15.8 30.6 13.0 15.3 29.0 3 15.5 20.2 44.5 18.4 25.0 47.8 19.4 24.5 52.5 5 18.8 26.6 56.1 22.8 32.5 69.1 24.5 36.0 78.2 10 23.8 36.6 72.1 29.8 47.8 93.4 32.6 54.0 107 Table 5. Secant deformation moduli Es [MPa] for piles in cohesionless soils. Layer thickness Moduli for given pile diameter and cohesive soil type [MPa] h d = 0.6 m d = 1.0 m d = 1.5 m [m] IC = 0.5 0.75 ≥ 1 IC = 0.5 0.75 ≥ 1 IC = 0.5 0.75 ≥ 1 1.5 6.9 10.0 13.2 7.9 10.7 13.4 8.6 10.5 12.3 3 10.0 15.5 22.0 12.5 18.6 23.9 13.7 18.4 23.0 5 12.5 21.9 31.2 15.9 25.7 33.4 18.4 27.6 36.7 10 15.5 29.9 44.3 21.3 36.3 51.3 24.6 41.0 57.4 Table 6. Secant deformation moduli Es [MPa] for piles in cohesive soils. with the influence factor of pile settlement: I = I1Rk, (12) consisting of the basic influence factor I 1 multiplied by the correction factor Rk representing the effect of the pile stiffness K = Epile Es and slenderness L d may be derived from plots presented in [31] or [55]. For layered soil deposits, the secant deformation modulus along the pile shaft is replaced by its weighted average: Es = n∑ i=1 Esihi n∑ i=1 hi , (13) using the moduli for individual soil layers in Tables 4, 5, and 6 determined from the back analysis of static loading tests. A.2. Refined approach providing a nonlinear load-settlement curve The nonlinear load-settlement curve is constructed according to the procedure outlined in Figure 14. First, the pile length is divided into N elements (Figure 5), and a small vertical displacement (settlement s0) is assigned to the pile base. Then, the pile base resistance q0 caused by its movement s0 is derived from a transfer function. For the purpose of this study, which focuses on negative skin friction, we adjusted this relationship to achieve the same pile resistance as in the previous practical approach because we wanted to separate the errors resulting from a different approximation of the base response. Then, the resultant force acting on the pile base with its cross-sectional area A0 follows as R0 = q0A0. For the shaft, we employed the load transfer functions motivated by Ménard’s rheological model, described in [55]. Then, the shaft friction for the i-th element qsi is supposed to be related to the vertical movement of 602 vol. 64 no. 6/2024 Settlement of piles with negative skin friction Set s0 = 1 mm s0 = s0 + 1s0 > smax Set i = 1 Find q0 for s0 R0 = q0A0 si−1 = s0 Find qsi for si−1 Ri = Ri−1 + qsiπdili Evaluate ∆li si = si−1 + ∆li Exit YES NO i = i + 1 i > N YES NO Figure 14. Flowchart of the algorithm providing the nonlinear load-settlement curve. Rock/Soil αi fi Ep,i [kPa] Weak rocks with uniaxial compressive strength of 15–20 MPa (R3) 0.66 3.02 80 Weak rocks with uniaxial compressive strength of 5–15 MPa (R4) 0.66 3.02 50 Weak rocks with uniaxial compressive strength of 1.5–5 MPa (R5) 0.66 3.02 30 Cohesionless soils with density index ID = 0.9 (D9) 0.66 3.02 40 Cohesionless soils with density index ID = 0.7 (D7) 0.66 3.02 20 Cohesionless soils with density index ID = 0.5 (D5) 0.66 3.02 10 Cohesive soils with consistency index IC ≥ 1.0 (C100) 0.5 4.50 25 Cohesive soils with consistency index IC = 0.75 (C075) 0.5 4.50 15 Cohesive soils with consistency index IC = 0.5 (C050) 0.5 4.50 8 Table 7. Parameters αi fi, and pressiometric moduli Ep,i for various rocks and soils. the element s(i−1) (see Figure 5) by: qsi = 0.7mqs,lim,i ( 1 − ( 1 − s(i−1) ss,lim,i )fi ) . (14) Here, the parameter m follows again from Table 2, the limit skin friction qs,lim,i is derived from the regression values following the previous approach, and the corresponding pile head settlement is assumed to be: ss,lim,i = 0.7qs,lim,i Ep gi(αi, d). (15) Other required parameters for the approximation of the skin friction mobilisation are listed for different types of rocks and soils in Table 7 and complemented with the function g(αi, di) (Table 8). Due to the fine division of the pile into individual parts, we assume that the average vertical movement of the element si−1 corresponds simply to the vertical settlement of its bottom plane. Then, the force acting on the top cross-section (Figure 5) holds: Ri = Ri−1 + qsiπdili. (16) Using the elastic contraction of the element length: ∆li = (Ri + Ri+1)li 2EpileAi , (17) 603 A. Zemanová, V. Anderle, J. Masopust Acta Polytechnica Diameter αi = 0.5 αi = 0.66 di [m] 0.6 2.997 1.892 0.8 3.460 2.287 1.0 3.869 2.650 1.2 4.238 2.989 1.4 4.577 3.309 1.6 4.893 3.614 1.8 5.190 3.906 2.0 5.471 4.188 Table 8. Values of function gi for different pile diameters di and rheological coefficients αi. the settlement of the element top is written as: si = si−1 + ∆li. (18) In this way, one proceeds to the pile head, where its settlement is derived. The force at the pile head RN corresponds to the pile resistance for the given base and head settlement s0 and sN . Repeating this procedure for gradually increasing base settlement (Figure 14) provides couples of pile head forces and settlements that can be plotted, providing the nonlinear load-settlement curve. 604 Acta Polytechnica 64(6):588–604, 2024 1 Introduction 1.1 Motivation 1.2 Literature review 1.3 Negative skin friction in current European standards 1.4 Goal of the study 2 Methods for pile-settlement and negative skin friction consideration 2.1 Quadratic-linear load-settlement curve approximation 2.2 Procedure providing nonlinear load-settlement curve 3 Examples 4 Results and discussion 4.1 Example 1 4.2 Example 2 5 Conclusion Acknowledgements References A A.1 Practical approach providing a quadratic-linear approximation of the load-settlement curve A.2 Refined approach providing a nonlinear load-settlement curve